2024/12/26 by Marco Bertola, Bertola, Marco, Alexander Tovbis +1 · 1 citation
Engineering · Physics and Astronomy · #Analysis of PDEs (math.AP) #Cold Atom Physics and Bose-Einstein Condensates #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Particle accelerators and beam dynamics #Strong Light-Matter Interactions
paper · pdf · doi:10.48550/arxiv.2412.19373
openalex publication_date 2024/12/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the family of (poly)continua \K in the upper half-plane \mathbb H that contain a preassigned finite \it anchor set E∈\mathbb H. For a given harmonic external field we define a Dirichlet energy functional \mathcal I(\mathcal K) and show that within each ``connectivity class'' of the family, there exists a minimizing compact \mathcal K^* consisting of critical trajectories of a quadratic differential. In many cases this quadratic differential coincides with the square of the real normalized quasimomentum differential \rm d \bf p associated with the finite gap solutions of the focusing Nonlinear Schrödinger equation (fNLS) defined by a hyperelliptic Riemann surface \mathfrak R branched at the points E∪ E. The motivation for this work lies in the problem of soliton condensate of least average intensity such that a given anchor set E belongs to the poly-continuum \mathcal K. An fNLS soliton condensate is defined by a compact \mathcal K⊂\mathbb H (its spectral support) whereas the average intensity of the condensate is proportional to \mathcal I(\mathcal K). We prove that the spectral support \mathcal K^* provides the fNLS soliton condensate of the least average intensity within a given ``connectivity class''.